Looking at the Remington web site ballistics, comparing the 308 Win and the 300 WSM, both with 180gr bullets. It's not apples to apples, cause they use a premium boat tailed bullet in the WSM and flat base in the 308.
The 308 is 2.3" high at 100 and zeroed at 200. It's then 3.8" low at 250 and 9.5" low at 300. It retains 1601 ft lbs of energy at 300yds.
The 300 WSM with the premier accutip is 2.5" high at 100, and zeroed at 250. At 300, it's 3.4" low. It retains 2378 ftlbs of energy, big gain there ,but is it and the associated recoil needed?
But in fairness, looking at the 308 with a 165gr accutip boattail bullet, it's still 8.6" low at 300, so maybe the comparison is fair. Given that, 6" difference at 300 yards isn't a whole lot in my book, especially on an elk sized animal.
In short, I, too, agree with GB, get the short throw, 308 Win. It would be MY choice in a BLR. A 165gr core lokt bullet would be great for the deer and bear, and go up to the 180 for elk.